2013 AMC 8 Problems/Problem 23
Contents
Problem
Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length . What is the radius of the semicircle on ?
Video Solution for Problems 21-25
Video Solution
https://youtu.be/crR3uNwKjk0 ~savannahsolver
Solution 1
If the semicircle on were a full circle, the area would be .
, therefore the diameter of the first circle is .
The arc of the largest semicircle is , so if it were a full circle, the circumference would be . So the .
By the Pythagorean theorem, the other side has length , so the radius is
Solution 2
We go as in Solution 1, finding the diameter of the circle on and . Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is , and the middle one is , so the radius is .
Video Solution by OmegaLearn
https://youtu.be/abSgjn4Qs34?t=2584
~ pi_is_3.14
See Also
2013 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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